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When dealing with integrals and Riemann sums, a common approach is to approximate the area under a curve using a method known as the Left-Endpoint Sum. This method involves using the left endpoints of subintervals to calculate the approximate area under the curve.To start with, you first divide the interval \[a, b\] into smaller equally spaced subintervals. Then, you calculate the width of each subinterval, which is also known as \( \Delta x \).Next, for each subinterval, you take the function value at the left endpoint. This value represents the height of the rectangle over that subinterval.Finally, you multiply each function value (at the left endpoints) by the subinterval width \( \Delta x \). Use these products to sum up the areas of all the rectangles and obtain the Left-Endpoint Sum. This is typically used to estimate integrals when finding the exact area is too complicated or impossible.Using the Left-Endpoint Sum is particularly helpful when the function is decreasing, as it tends to give a more conservative approximation. However, remember that increasing functions might yield underestimations in the area using this method.

The Right-Endpoint Sum is another method used to approximate the integral of a function by partitioning its interval into equal subintervals and using the right endpoints to estimate the area under the curve.Begin by identifying the main interval \[a, b\] and dividing it into smaller subintervals. The subinterval width is computed using the formula \( \Delta x = \frac{b-a}{n} \), where \( n \) is the number of subintervals.Then, for each subinterval, evaluate the function at the right endpoint. This function value represents the rectangle's height over that subinterval.Finally, multiply each evaluated function value by the subinterval width \( \Delta x \) and then sum up all of these products. The total gives you the Right-Endpoint Sum, which approximates the area under the curve.The Right-Endpoint Sum often provides a better approximation for increasing functions, as it tends to overestimate the area, while it might underestimate for decreasing functions. This method complements the Left-Endpoint Sum, providing a comprehensive approach to understanding area approximation.

The calculation of subinterval width is a crucial step in performing Riemann Sum approximations, either with left or right endpoint methods.For any interval \[a, b\], the width \( \Delta x \) of each subinterval can be calculated easily. The formula \( \Delta x = \frac{b-a}{n} \) determines this, where:

• \( a \) and \( b \) are the endpoints of the interval.
• \( n \) is the number of subintervals you are dividing the interval into.

This calculated width, \( \Delta x \), is the base length of each rectangle used to approximate the integral's area under a curve. Consistent subintervals (same \( \Delta x \) for each), help in maintaining uniformity which simplifies calculations, especially when finding sums like the Left-Endpoint or Right-Endpoint Sum.Applying the formula ensures that each subinterval is of the same size, which contributes to better accuracy in the approximation of integrals and provides a consistent structure in solving such problems. It's always a basic step but one that forms the foundation of further calculations and understanding of Riemann Sums.